Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $r = \dfrac{-7t - 42}{4t + 36} \div \dfrac{t^2 + 10t + 24}{3t + 27} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{-7t - 42}{4t + 36} \times \dfrac{3t + 27}{t^2 + 10t + 24} $ First factor the quadratic. $r = \dfrac{-7t - 42}{4t + 36} \times \dfrac{3t + 27}{(t + 6)(t + 4)} $ Then factor out any other terms. $r = \dfrac{-7(t + 6)}{4(t + 9)} \times \dfrac{3(t + 9)}{(t + 6)(t + 4)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ -7(t + 6) \times 3(t + 9) } { 4(t + 9) \times (t + 6)(t + 4) } $ $r = \dfrac{ -21(t + 6)(t + 9)}{ 4(t + 9)(t + 6)(t + 4)} $ Notice that $(t + 9)$ and $(t + 6)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ -21\cancel{(t + 6)}(t + 9)}{ 4(t + 9)\cancel{(t + 6)}(t + 4)} $ We are dividing by $t + 6$ , so $t + 6 \neq 0$ Therefore, $t \neq -6$ $r = \dfrac{ -21\cancel{(t + 6)}\cancel{(t + 9)}}{ 4\cancel{(t + 9)}\cancel{(t + 6)}(t + 4)} $ We are dividing by $t + 9$ , so $t + 9 \neq 0$ Therefore, $t \neq -9$ $r = \dfrac{-21}{4(t + 4)} ; \space t \neq -6 ; \space t \neq -9 $